Fig. 1. Geometry and axes definition of an AUV.

It is convenient to write the equations of motion in accordance with the Society of National Architects and Marine Engineers (SNAME, 1950). Restricting our attention to the horizontal plane, the mathematical model consists of the nonlinear sway (translational motion with respect to the vehicle longitudinal axis) and yaw (rotational motion with respect to the vertical axis) equations of motion. According to (Haghi et al., 2007), these equations are described by v[m -YJ + r[mxG -Y r ] = Y sSp2 +Y sSbu2 - d1(v , r ) +Yvuv + (Y r - m )ur (19)

v[mxG -N^] + r[Iz -Nr] = Ns Ssu2 + NsSbu2 -d2(v ,r) + Nvuv + (Nr - mxG)ur, (20) where dj(v ,r) and d2(v ,r) are defined as di(v >r) = I,., CDyh(4)--—d4

Equations (19) and (20), along with the expressions for the vehicle yaw rate and the inertial position rates, describe the complete model of the vehicle. For control purposes it is convenient to solve Eqs. (19) and (20) for v and r . Therefore the complete set of equations of motion is v = a11uv + a12ur + dv (v, r) + b11u 2Ss + b12u 2Sb (21)

r = a21uv + a22ur + dr (v ,r) + b21u 2Ss + b22u 2Sb (22)

where atJ , btJ and ci are the related coefficients that appear when solving (19) and (20) for v and r .

During regular cruising, the drag related terms dv (v , r ) and dr (v, r ) are small, and can be neglected (Yuh, 1995). Note that all the parameters atJ and btJ , include at least two hydrodynamic coefficients, such as YvYr,Nv,N r ,... ; hence uncertainties. In the proceeding sections, we apply the nonlinear control methods of the previous section to this model. Our goal is to achieve perfect tracking for both sway and yaw motions of the vehicle.

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